Quadratic Number Fields

Imagine you have a sphere in R^3 with radius equal to the square root of n. How many points with integer coordinates are there on this sphere? Gauss gave a formula in Disquisitiones Arithmeticae relating this number to the structure of quadratic number fields. Generally, the rings of integers of quadratic fields do not have unique factorization into primes, and the failure of unique factorization is measured by the class number. Gauss related the number of integral points on the sphere to the class number.

 

Class numbers are connected to deep mathematics. Proving Gauss’ original conjectures for imaginary quadratic fields and making them effective took place hand-in-hand with many of the great developments of number theory throughout the twentieth century, including Siegel’s work on zeros of Dirichlet L-functions and work by Gross and Zagier, and Goldfeld on elliptic curves, via ideas that have been important in some of the best theorems to date on the Birch and Swinnerton-Dyer Conjecture.

 

There’s still a lot we don’t know about class numbers. I have worked on understanding the arithmetic properties of the class number. Predictions by Cohen and Lenstra describe many divisibility statistics, however we are a long way from proving their conjectures. I have studied divisibility questions for class numbers of imaginary quadratic fields, and some of their implications for the structure of elliptic curves.

Papers related to quadratic fields:

Congruences of Hurwitz class numbers on square classes, (with Martin Raum and Olav Richter). Submitted. Link.

 

Non-holomorphic Ramanujan-type congruences for Hurwitz class numbers (with Martin Raum and Olav Richter). Proc. Natl. Acad. Sci. USA (PNAS), 117 (2020), no. 36, 21953-21961. Link.

 

Class number divisibility for imaginary quadratic fields.  Research in Number Theory,   Res. Number Theory 6 (2020), no. 1, Paper No. 13.  Link. 

 

Indivisibility of class numbers of imaginary quadratic fields. Res. Math. Sci. 4 20  (2017). Link.

Integer Partitions

Say you want to know how many ways you can write n as a sum of positive integers, up to reordering the summands. The number of ways is the nth partition number p(n). A hundred years ago, Ramanujan discovered three surprising divisibility properties for the sequence p(n): he discovered that p(5n+4) is always divisible by 5, p(7n+5) is always divisible by 7, and p(11n+6) is always divisible by 11. Trying to understand these and similar congruences led to important work on newforms by Atkin and Lehner and we now know that many similar congruences come about from the theory of Galois representations attached to modular forms.

 

If we put the partition numbers into a generating function, the resulting function has lots of symmetries and is a modular form. Using the theory of modular forms, I have studied integer partitions both in terms of their asymptotic properties and arithmetic properties, including work towards understanding the Ramanujan-type congruences that can occur.

 

One question I’m interested in is classifying the types of Ramanujan-type congruences that can occur, more specifically, finding a description for the triples (A,B,p) such that p(An+B)=0(mod p) for all n. For example, are there any of the form (Q*P, B, P), where Q, P are primes, and P is at least 13? This is not known, but my work with Ahlgren and Raum suggests that such triples, if they exist, are rare in a certain sense. On the other hand, several students I supervised at Tulane and the Illinois Geometry Lab have found that such congruences do exist for colored partitions.

 

Papers related to partitions: 

Scarcity of congruences for the partition function (with Scott Ahlgren and Martin Raum). Accepted by American Journal of Mathematics. Link. 

On the number of parts of integer partitions lying in given residue classes (with Michael Mertens). Annals of Combinatorics 21 (2017), p. 507-517. Link. 

The number of parts in certain residue classes of integer partitions (with Michael Mertens). Research in Number Theory  1 (2015). Link. 

Multiplicative Properties of the Number of k-Regular Partitions (with Christine Bessenrodt). Annals of Combinatorics 20 (2016), p. 231-250. Link.

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Applications in the sciences

I work in some of the more abstract areas of math, but I also seek out situations where the ideas can be put to use in other fields. 

A lot of exciting high-brow math has been developed from trying to understand the partition numbers. On the other hand, counting is so ubiquitous in science that the partition numbers turn out to appear in chemistry and physics too, particularly in counting states of physical systems. Fellow number theorist Robert Schneider and I co-wrote a paper with the Kindt Research Group in the Emory University Department of Chemistry. The mathematics formed the basis of a robust and efficient algorithm for modeling molecular aggregation.

Extracting aggregation free energies of mixed clusters from simulations of small systems (with Xiaokun Zhang, Lara Patel, Robert Schneider, C.J. Weeden, and James Kindt).  J. Chem. Theor. Comp. 13, p. 5195-5206 (2017).  Link. 

Modular forms and L-functions

The main tools in my research are elliptic modular forms. These are analytic functions on the hyperbolic plane with nice symmetry. Their Fourier coefficients encode arithmetic information, such as the sum-of-divisors functions, they have a rich theory of linear operators, elegant properties at quadratic irrationalities, and important applications in arithmetic geometry. 

Mellin transforms of modular forms are L-functions, a class of analytic functions in number theory which include the Riemann zeta function and Dirichlet L-functions. The distribution of zeros of L-functions encode arithmetic information such as the distribution of primes and coefficients of modular forms. For example, the most famous open problem in number theory is the Generalized Riemann Hypothesis, which predicts that all the nontrivial zeros of L-functions have real part equal to ½. Knowing GRH for the Riemann zeta function would give us a very strong approximation for the prime counting function. With Liu, Thorner, and Zaharescu, I have been studying statistical approximations to GRH. As an application, we've explored the distributions of the fractional parts of the zeros of a class of L-functions. 

A zero density estimate and fractional imaginary parts of zeros for GL(2) L-functions (with D. Liu, J. Thorner, and A. Zaharescu). Submitted. Link.

Bounds for special values of shifted convolution Dirichlet series. Proc. Amer. Math. Soc. 145 6 (2017). p. 2373-2381. Link.